Dynamic Characteristic Analysis of Underwater Suspended Docking Station for Resident UUVs

Abstract

The widespread use of Unmanned Underwater Vehicles (UUVs) in seafloor observatory networks highlights the need for docking stations to facilitate rapid recharging and effective data transfer. Floating docks are promising due to their flexibility, ease of deployment, and recoverability. To enhance understanding and optimize UUV docking with floating docks, we employ dynamic fluid body interaction (DFBI) to construct a seabed moored suspended dock (SMSD) model that features a guiding funnel, a suspended body, and a catenary of a mooring chain. This model simulates SMSD equilibrium stabilization in various ocean currents. Then, a UUV docking model with contact coupling is developed from the SMSD model to simulate the dynamic contact response during docking. The accuracy of the docking model was validated using previous experimental data. Through investigation of the UUV docking response results, sensitivity studies relating to volume, moment of inertia, mass, and catenary stiffness were conducted, thereby guiding SMSD optimization. Finally, sea tests demonstrated that the SMSD maintained stability before docking. During docking, the SMSD’s rotation facilitated smooth UUV entry. After the UUV docked, the SMSD was restored to its original azimuth, confirming its adaptability, stability, and reliability.

1. Introduction

Unmanned Underwater Vehicles (UUVs) are crucial in ocean exploration and development due to their autonomous capabilities [1]. They are used in rescue missions, pipeline inspections, marine science studies, the exploration of seabed energy, and fishing. However, their limited energy supplies restrict their operations, necessitating periodic retrieval by a mother ship, which is time-consuming and labor-intensive. Advances in seafloor observatory networks have introduced docking stations [2,3] where UUVs can recharge, upload data, and download missions, reducing the need for frequent retrievals. This innovation extends the operational range of UUVs and improves their autonomous potential [4]. As underwater tasks become more complex, the incorporation of UUVs into seafloor networks is increasing, making docking stations as essential as refueling or charging stations on land.

The variety of UUVs has resulted in the following three main docking configurations [5]: a funnel-shaped or cone-shaped entrance [6], a vertical design with a V-shaped capture mechanism [7], and a docking platform [8]. The cone-shaped entrance is used most frequently because it requires minimal modifications to the UUV and does not require additional mechanical parts [9]. This design is especially effective for retrieving torpedo-shaped UUVs and usually provides a large cross-sectional docking area.

Docking stations shaped like funnels or cones can be categorized into the following three main types [10]: fixed, mobile, and floating stations. Fixed docks are securely mounted to the seabed or underwater stations, providing high stability but necessitating flat seabeds. They may include self-adjusting mechanisms to adapt to uneven terrain. Examples are the Dorado AUV station [6] and the Dolphin II AUV station, which features a two-axis swing [11]. Innovations such as motor-driven rotation are used to counteract cross-flow effects during docking, as seen with the Hybrid Underwater Glider (HUG) single-axis rotating station [12] and the hydraulically adjustable station [13]. Although highly stable, fixed docks are complex, susceptible to marine biofouling, and difficult to deploy in the long term [14]. Mobile docks are attached to larger UUVs [15,16] or unmanned boats [17], using towing speed to ensure stable positioning. These docks are mainly designed for short-term recovery operations. Lastly, floating docks can either hang freely or be suspended at a specific height. They can be incorporated into buoy systems [18], surface ships, or seabed mooring systems [19], maintaining stability under ocean currents. Although simple in structure and adaptive to ocean currents, they can sway due to surface waves [20]. Their stability can be improved by anchoring them to the seabed.

A suspended seabed docking station is easy to deploy and retrieve, is not affected by seabed flatness, and is effective for UUVs to dock in diverse marine environments. As shown in Figure 1, the underwater local resident observation network around offshore platforms includes a suspended seabed dock, UUVs, and underwater wireless nodes for ocean monitoring and identification of equipment problems and safety risks. This paper focuses mainly on a suspended seabed dock (marked within the red box in Figure 1). It adopts a mooring type that links the dock to the counterweight through a mooring chain or a load-bearing cable. The design includes a suspended body structure that offers buoyancy. In addition, it can be connected to the seafloor junction box with a cable or remain unconnected (orange dashed line in Figure 1). Furthermore, the docking function can be extended to wireless nodes to form dock nodes, allowing UUVs to power them, prolonging the operating time of the monitoring network [21]. The design allows UUVs and Remotely Operated Vehicles (ROVs) to operate independently of established subsea resources [22]. The flexible design of suspended stations uses ocean currents for passive posture adjustments, enhancing stability and adaptability in dynamic environments. As researchers seek efficiency, stability, and reliability, there is a strong need to implement adaptive docking stations and explore their potential for performance.

As a novel structural form, research on underwater suspended docking stations is currently limited. Ensuring the success and safety of UUV docking operations requires attention to the station’s static stability and dynamic contact reliability. First, static stability is essential for accurate positioning and safe docking of UUVs. R. Zheng et al. examined the impact of wing size, tow cable length, and hinge position on the motion state of a suspended dock with mooring [23]. X. Wen et al. used a numerical method and a modified P controller with an added pre-condition and limiter to assess the dynamic response of a floating dock in accidents, enhancing its safety and stability [24]. Stability analysis often uses Computational Fluid Dynamics (CFD) for tow-mobile docks, examining hydrodynamic coefficients, tow cable characteristics, and tow motion parameters [25]. Edoardo I. et al. simulated the automated launch and recovery process using OrcaFlex and performed a parametric study [17]. Furthermore, studies on wave energy converters [26] and floating offshore wind turbines [27,28] provide information on the numerical modeling, dynamic performance, and survivability of moored floating platforms.

Contact reliability is crucial for docking efficiency and success, with the aim of minimizing contact forces and using them to guide the UUV smoothly into the dock [29]. Water flow disturbances often cause UUVs to collide at the dock entrance, increasing difficulty and the risk of damage. Frequent problems involve the UUV becoming lodged at the entrance [30], damage to the guide cone, or deformation of the UUV’s front. B. Fletcher et al. suggested using water-filled bags to increase and stabilize suspended dock mass. They highlighted the need for the dock to offer adequate reaction forces and torque (from inertia and/or thrust) to ensure that the UUV’s docking efficiency is not affected by the dock’s movements [31,32]. M. Lin et al. developed an ADAMS contact model to study floating docks, analyzing factors such as the cable length and mass ratio [30]. In the context of fixed docks, Zhang et al. created a contact model to examine the impact of the guide cone material, the thrust of the Autonomous Underwater Vehicle (AUV), and the initial position [33]. Our previous work evaluated the effects of different guide-cone shapes (convex, conic, and concave) on the adjustment of UUV motion [34]. Other researchers have explored optimizing the shape of contact bodies and using multi-objective optimization to reduce contact forces. Furthermore, Wu Lihong was the first to use CFD techniques to analyze the hydrodynamic characteristics of AUV docking, taking into account factors such as speed, acceleration, entrance geometry, sliding behaviors, and rudder angles [35,36]. Meng Lingshuai used Star-CCM+ to gather contact force data from captured rod docking simulations [29], and Xu Yunxin investigated the hydrodynamic feasibility of dynamically docking a remora-inspired AUV on a benchmark submarine [37]. A novel “soft dock” design featuring flexible appendages and active gripping mechanisms can significantly reduce collision impacts [38]. Despite the advances in the field, research on the hydrodynamics of UUV docking with suspended docks is still lacking, particularly in terms of fully understanding the post-contact motion response crucial for successful docking. Hence, further studies are essential to utilize underwater fluid dynamics and contact interactions more effectively to enhance UUV docking, focusing specifically on post-contact motion.

This paper presents a Seabed Moored Suspended Dock (SMSD) model and a UUV docking model with contact coupling to simulate SMSD stabilization under various currents and the UUV’s dynamic contact response during docking. The goal is to deepen the understanding of the SMSD’s motion response to water flow and contact forces to optimize UUV docking, providing design insights for floating docks. The SMSD is composed of a guiding funnel, a suspended body, and a catenary. To realize six-degree-of-freedom (6-DOF) motion for both the UUV and SMSD, techniques such as Dynamic Fluid Body Interaction (DFBI), overlapping grids, and adaptive mesh refinement are employed. The accuracy of the UUV docking model is validated using experimental data. Then, by systematically analyzing the effects and sensitivities of the design parameters on the attitude of the SMSD and the contact response during docking, we provide suggestions for optimizing floating dock designs. Finally, based on the proposal for optimal parameters, we construct the SMSD and perform sea tests to confirm the stability and contact reliability of its suspended moored design.

The main contributions of this work are summarized as follows.

• This study proposes an SMSD model using a hybrid dynamic overlapping grid technique through DFBI, featuring a guiding funnel, a suspended body, and a catenary. Moreover, a UUV dock within the SMSD model is developed with contact coupling. These models successfully simulate the SMSD’s equilibrium stability and dynamic contact response under ocean flow conditions.
• We analyze the effects of mass, catenary stiffness, and flow velocity on SMSD stability. In addition, an investigation is conducted regarding how volume, moment of inertia, mass, and catenary stiffness impact UUV docking. This improves understanding of flexible mooring methods and contact interactions, guiding the optimization of SMSD design.
• Through sea trials and numerical simulation results, we verify the stability and contact reliability of the suspended moored structure of the SMSD. The well-considered design enhances the SMSD’s adaptability to environmental conditions and stability during UUV docking.

This paper is structured as follows. Section 2 details the numerical methods used for simulation of the attitude of the SMSD and the UUV docking within the SMSD, including the catenary equations, contact coupling and forces, and the DFBI motion. Section 3 outlines the calculation model, grid division, and validation of the numerical model. Section 4 analyzes SMSD equilibrium stability and contact interaction during UUV docking, as well as the impacts of volume, moment of inertia, mass, and catenary stiffness. Section 5 presents the sea experiments and results of the optimized SMSD. Finally, Section 6 summarizes the findings of this study and identifies deficiencies and directions for future work.

2. Numerical Methodology

UUV docking is a complex task that involves hydrodynamic, structural, and multibody interactions, with both the UUV and the docking station affected by fluid disturbances. The hydrodynamic model utilizes the Finite Volume Method (FVM) implemented in STAR-CCM + software [27,28] to solve the Reynolds Average Navier–Stokes (RANS) equation, simulating the fluid dynamics around the SMSD and the UUV. This research adopts the K-Epsilon turbulence model [29,39].

At the moment of contact, the influence of the liquid is minimal and, therefore, can be ignored [40]. As a result, this research focuses on the motion responses of the SMSD and UUV after contact, as affected by hydrodynamic forces. To simplify the problem, we assume the following:

(1)

Water flow is treated as a constant disturbance;

(2)

The propulsion of the UUV is modeled to provide a steady thrust at a constant rotational speed;

(3)

Both the UUV and SMSD are assumed to be rigid bodies, without deformation occurring during the contact process;

(4)

The duration of contact is fairly short and does not cause infinite interpenetration in the contact region [33]. The bodies pass through each other (infinite interpenetration) [41].

2.1. Catenary Equations

The mooring chain is described by catenary coupling. This method models an elastic, quasi-static catenary (like a chain or towing rope) suspended between two end points and influenced by its own weight in a gravitational field. In a local Cartesian coordinate system, the shape of the catenary is given by the following parametric equations [42]:

$$\left\{\begin{array}{c}x=au+bsinh\left(u\right)+\alpha \\ y=acosh\left(u\right)+{\displaystyle \frac{b}{2}}{sinh}^2\left(u\right)+\beta \\ {foru}_1\le u\le u_2\end{array}\right.$$

(1)

where $\alpha$ and $\beta$ are integration constants depending on the position of the two end points and the total mass of the catenary, $u_1$ and $u_2$ represent the positions of the catenary’s end points ($p_1$ and $p_2$, respectively) in parameter space. The definitions of a and b are

$$\left\{\begin{array}{c}a={\displaystyle \frac{c}{{\lambda }_{0}g}}\\ b={\displaystyle \frac{ca}{D{L}_{eq}}}\\ c={\displaystyle \frac{{\lambda }_0{L}_{eq}g}{sinh\left(u_2\right)-sinh\left(u_1\right)}}\end{array}\right.$$

(2)

where g is the gravitational acceleration; ${\lambda }_0$ and $L_{eq}$ are the mass per unit length and the relaxation length of the catenary under force-free conditions, respectively; and D is the stiffness of the catenary.

The curve parameter (u) is related to the inclination angle ($\varphi$) of the catenary curve by the following equation:

$$tan\varphi =sinh\left(u\right)$$

(3)

The force (${\mathbf{T}}_1$) acting at one end point of the catenary on the SMSD is directed along the local tangent vector of the catenary curve at the parameter value of $u_1$. It is given by the following expressions:

$$\left\{\begin{array}{c}{\mathbf{T}}_{1,x}=c\\ {\mathbf{T}}_{1,y}=ccosh\left(u_1\right)\end{array}\right.$$

(4)

2.2. Contact Coupling and Forces

To accurately reflect the interaction between the UUV and the SMSD, we employ the contact coupling method to simulate the contact force [43]. To prevent contact, the model applies a contact force that depends on the distance between the rigid body boundary and the opposing boundary. The contact force decreases to zero when the distance between the body boundary and the other boundary is larger than a user-specified effective range. If the distance is smaller than the effective range, a repulsive contact force is applied [42]. The contact force comprises the following two components: the normal component, which prevents penetration, and the tangential component, which models friction as the body slides along the opposing boundary.

The normal component of the contact force can be written as follows:

$${\mathbf{F}}_{c,n}\left(d_f\right)=a_f\left[k_1(d_0-d_f)-k_2\dot{d_f}\right]{\mathbf{n}}_{bf}$$

(5)

where $k_1$ is the elastic coefficient, $k_2$ is the damping coefficient, $a_f$ is the area of the contact face mesh, $d_f$ is the distance between the face centroid and the opposing boundary, $d_0$ is the effective range, and ${\mathbf{n}}_{bf}$ is the normal direction of the boundary face next to the body face (f).

The contact force consists of elastic and damping components determined by the elastic coefficient ($k_1$) and the damping coefficient ($k_2$), respectively. The elastic coefficient ($k_1$) is essential to stop the rigid body before contact is made. It is estimated as follows:

$$k_1={\displaystyle \frac{m_u{m}_d}{m_u+m_d}}·{\displaystyle \frac{1}{A}}·{\displaystyle \frac{v_{n,rel}^2}{{(d_0-d_{min})}^2}}$$

(6)

where $m_u$ and $m_d$ denote the masses of the UUV and SMSD, respectively; A is the estimated area of contact; $v_{n,rel}$ is the normal relative impact velocity of the UUV and SMSD; and $d_{min}$ is the minimum distance. The UUV should stop before this distance is reached.

The damping coefficient ($k_2$) is estimated as follows:

$$k_2=2\zeta \sqrt{{\displaystyle \frac{{(m}_u·m_d)k_1}{(m_u+m_d)A}}}$$

(7)

where $\zeta$ is a constant factor describing the amount of damping. In general, $\zeta$ should be sufficiently small ($\zeta \ll 1$), as large values of $\zeta$ in combination with a large time step can cause numerical instability.

The tangential component of the contact force on face f is given by

$${\mathbf{F}}_{c,t}=-\mu \left|{\mathbf{F}}_{c,n}\right|tanh\left(k_t{\mathbf{v}}_t\right)$$

(8)

where $\mu$ is the friction coefficient, $k_t$ is the tanh coefficient (using the default value), and ${\mathbf{v}}_t$ is the tangential velocity of face f relative to the opposing boundary. The tanh function makes the friction force stable for low relative sliding velocities.

The contact force on boundary face f can be written as follows:

$${\mathbf{F}}_c={\mathbf{F}}_{c,n}+{\mathbf{F}}_{c,t}$$

(9)

The contact force results in a moment (${\mathbf{M}}_c$) around the current body position (${\mathbf{r}}_b$) as follows:

$${\mathbf{M}}_c=({\mathbf{r}}_f-{\mathbf{r}}_b)\times {\mathbf{F}}_c$$

(10)

where ${\mathbf{r}}_f$ is the location of the face centroid of face f.

2.3. DFBI Motion

The DFBI solver is capable of simulating the motion of the rigid body caused by forces of the physical continuum, calculating the resulting forces and torques in the rigid body and solving the motion control equations to find the instantaneous position of the body [27]. As shown in Figure 2, the DFBI rotation and translation method addresses the 6-DOF for SMSD and UUV. The origins of moving coordinate systems D-$x_d{y}_d{z}_d$ and U-$x_u{y}_u{z}_u$ are located in the center of mass of the SMSD and UUV, respectively.

For instance, considering the SMSD, the translational motion equation of the center of mass is formulated in the global inertial coordinate system as follows:

$$m_d{\displaystyle \frac{d\mathbf{v}}{dt}}={\mathbf{S}}_g+{\mathbf{F}}_f+{\mathbf{T}}_1+{\mathbf{F}}_c$$

(11)

where $m_d$ is the mass of the SMSD, ${\mathbf{S}}_g$ is the net buoyancy force, ${\mathbf{F}}_f$ is the fluid force, ${\mathbf{T}}_1$ is the catenary force, ${\mathbf{F}}_c$ is the contact force, and $\mathbf{v}$ is the velocity of the center of mass of the SMSD.

The equation of rotation of the SMSD is formulated in the body local coordinate system as follows:

$$\mathbf{M}{\displaystyle \frac{d\mathbf{\omega }}{dt}}+\mathbf{\omega }\times \mathbf{M}\mathbf{\omega }={\mathbf{M}}_G+{\mathbf{M}}_f+{\mathbf{M}}_T+{\mathbf{M}}_c$$

(12)

where $\mathbf{M}$ is the tensor of the moments of inertia, $\mathbf{\omega }$ is the angular velocity of the SMSD, ${\mathbf{M}}_G$ is the gravitational moment, ${\mathbf{M}}_f$ is the fluid moment, ${\mathbf{M}}_T$ is the catenary moment, and ${\mathbf{M}}_c$ is the contact moment.

The motion equation for the UUV is similar to that of the SMSD. However, in the external force, the tension of the mooring chain ${\mathbf{T}}_1$ is replaced by $\mathbf{T}$, which represents the main thrust that acts on the UUV. The external moment follows a similar pattern. The expressions are as follows:

$$\left\{\begin{array}{c}{m}_u{\displaystyle \frac{d{\mathbf{v}}^{\prime }}{dt}}={{\mathbf{S}}^{\prime }}_g+{{\mathbf{F}}^{\prime }}_f+\mathbf{T}+{{\mathbf{F}}^{\prime }}_c\\ {\mathbf{M}}_u{\displaystyle \frac{d{\mathbf{\omega }}^{\prime }}{dt}}+{\mathbf{\omega }}^{\prime }\times {\mathbf{M}}_u{\mathbf{\omega }}^{\prime }={{\mathbf{M}}^{\prime }}_G+{{\mathbf{M}}^{\prime }}_f+{{\mathbf{M}}^{\prime }}_T+{{\mathbf{M}}^{\prime }}_c\end{array}\right.$$

(13)

where ${\mathbf{M}}_u$ is the tensor of the moments of inertia and ${\mathbf{\omega }}^{\prime }$ is the angular velocity of the UUV.

3. Numerical Model

In this section, we present the calculation models for the SMSD and the passive contact of the UUV with the SMSD, describing the structural parameters, boundary conditions, and dynamic parameters. After establishing the model, we perform grid division, assess grid independence, and validate the accuracy of the numerical simulation by comparing it with experimental results.

3.1. Calculation Model

As illustrated in Figure 3, two computational models are developed—one for the attitude of the SMSD and another for the UUV docking into it, with both considered to be rigid bodies. As shown in Figure 3b, the SMSD consists of a guiding funnel, a suspended body, and a catenary. The guiding funnel is firmly connected to the suspended body, creating a unified structure anchored to the seabed by the catenary. The initial configuration of the SMSD is detailed in

Table 1. Initial configuration of the SMSD.

Parameter	Notation	Value
Diameter of guiding funnel entrance	o	1.85 m
Angle of guiding funnel entrance	$\theta$	${51}^{\circ }$
Overall dimensions	$D_D$	2.886 × 2.674 × 4.678 m
Mass	$m_d$	2800 kg
Volume	$V_D$	3.55 m3
Moment of inertia	$M_I$	[2000, 2000, 1800] kg·m2
Length of the catenary	$L_{eq}$	3.1 m
Mass per unit length of the catenary	${\lambda }_0$	1.4 kg/m
Stiffness of the catenary	D	6,482,580 N/m
Velocity of flow	$v_f$	0.5 m/s

, with the parameters adjustable for different simulation scenarios. The catenary has a constant length, and its anchor point is located 0.5 m above the seabed. Additionally, the central axis of the guiding funnel is 4.2 m above the seabed. The suspended body provides buoyancy and stability, facilitated by a hexagonal frustum design. This evenly distributes buoyancy around the edges, improving stability by reducing the amplitude of slipping or rolling. The symmetrical shape of the SMSD assists in aligning the guiding funnel with the flow direction, facilitating the stable navigation of the UUV. In Figure 3c, $D_0=2.75\mathrm{m}$ is the distance between two opposite sides of the suspended body. The initial configurations for the UUV and the contact coupling are described in

Table 2. Initial configuration of UUV docking.

Parameter	Notation	Value
Overall dimensions	$D_U$	2.925 × 0.4 × 0.43 m
Mass	$m_u$	350 kg
Volume	$V_U$	0.56 m3
Moment of inertia	$M_i$	[14, 227, 230] kg·m2
Velocity	$v_u$	[0.6, 0, 0] m/s
Horizontal posture deviation	$\Delta \alpha$	${15}^{\circ }$
Vertical posture deviation	$\Delta \beta$	$0^{\circ }$
Horizontal position deviation	$\Delta d_y$	0.777 m
Vertical position deviation	$\Delta d_z$	0 m

and

Table 3. Initial configuration of the contact coupling.

Parameter	Notation	Value
Elastic coefficient	$k_1$	1.512 × 1011 Pa/m
Damping coefficient	$k_2$	7609 Pa·s/m
Friction coefficient	$\mu$	0.05

, respectively. According to [44,45], the computational domain is designated as $25D_0\times 20D_0\times 11D_0$, with the SMSD body located $10D_0$ from the velocity inlet. The bottom seabed is modeled as a boundary condition of the wall.

The DFBI rotational and translational motions involve two 6-DOF continua, i.e., the SMSD and the UUV. To ensure that the UUV reaches the guiding funnel at the same ground-relative speed before making contact, the UUV is provided with a primary thrust that adjusts to different initial flow velocities. The model also incorporates the following two types of body coupling: catenary and contact coupling. Catenary coupling employs a pretension mode. Contact coupling uses a centroid-to-centroid contact method, with an effective range of 0.001 m.

3.2. Grid Division

The overlapping grid technology enables the dynamic simulation of the SMSD and UUV. The computational domain is segmented into several overlapping sub-grids, allowing those surrounding the moving parts to coordinate with their movements. In addition, overlapping grid interfaces are created between the fluid domain and the solid domains of the SMSD and the UUV. As shown in Figure 4, the trimmed grid is used to efficiently generate a high-quality mesh for complex geometric surfaces [46]. Volume refinement control is executed between the background and overlapping regions to ensure a smooth transition; specifically, the contact region is refined to better capture unsteady contact behaviors. To improve solution precision at boundaries, surface grids for both the SMSD and UUV are reconstructed, and boundary layers are defined to conform to the requirements of the turbulence model. The boundary layer grids at the interface between the guiding funnel and the UUV bow are carefully matched to resolve the contact. Furthermore, the time step is adjusted on the basis of the UUV’s velocity to maintain the accuracy of the contact coupling and avoid errors between the grid’s active and inactive states during calculation.

CFD techniques are highly dependent on computational resources to address real-time problems. To avoid the impact of the density of the boundary layer grid and the transition region between the overlapping grids on the calculation results, a grid independence validation is performed, as shown in

Table 4. Grid-independent verification.

Grid Number (Millions)	Pitch of SMSD (∘)	Relative Error (%)	Grid Number (Millions)	Contact Force (N)	Relative Error (%)
2.73	6.11	7.86	2.62	35,467	8.57
2.93	5.63	1.78	3.07	34,715	6.59
3.60	5.53	0.36	4.27	32,428	−2.65
4.29	5.51	0.18	4.88	33,286	−0.90
4.53	5.50		6.96	33,584

. The relative error reflects the relative variation in the physical quantity corresponding to the number of adjacent large and small grids. To ensure the accuracy of the results and to avoid wasting computing resources, the grid count for the UUV docking process is set at 4.27 million, with the relative error maintained within 3%.

3.3. Numerical Model Validation

The accuracy of STAR-CCM+ in simulating static forces and DFBI motion on rigid bodies has been validated by numerous studies. These studies cover a wide range of applications, including the hydrodynamic characteristics of UUVs, unmanned boat towing, buoy dynamics, and bullets entering the water. In this context, to validate the UUV docking model, a similar model is constructed that replicates the previous sea trial scenario. The docking station and UUV parameters are consistent with those of the actual experiments. As shown in Figure 5a, the experimental docking station is simplified to include only the guiding funnel and is equipped with a 3D electronic compass. The station is suspended from the upper part by slings attached to its four corners. Finally, it is submerged at a depth of 2 m using a crane, with the UUV being guided acoustically to the station. In Figure 5b, the slings are described by four catenary couplings. The numerical simulation reproduces a contact position similar to that observed in the experiment.

Upon successful docking of the UUV, its velocity was monitored using a Doppler Velocity Log (DVL), and the docking station’s rotation was recorded. The numerical results were compared with the experimental results using the validation method proposed in [33], adjusting the simulation timeline to match the duration of the experiment. As depicted in Figure 6a, the timing of the velocity variations in the simulation is closely aligned with the experimental data, although a smoother velocity curve is shown in the simulation. Despite the fact that there are minor magnitude discrepancies due to differences in the model and initial conditions, overall trends are consistent. Figure 6b illustrates that the docking station’s yaw-angle changes match well between the simulation and the experiment, with the simulation showing a linear change due to the uniform catenary setup. In contrast, the experimental yaw angle fluctuates as a result of asymmetric torques from the non-uniform slings caused by manual operation. The simulation results closely replicate the experimental trends, validating the accuracy of the numerical model in simulating UUV docking maneuvers.

4. Results and Discussion

This section analyzes the numerical simulation results for SMSD attitude stabilization and the UUV docking process. The main focus is the pitch angle of the dock and its orientation relative to the direction of the water flow before the UUV approaches, as well as the motion response of the dock during docking. First, we investigate the key parameters of the SMSD mass and the catenary stiffness that influence the attitude of the SMSD and discuss their impact patterns. Secondly, we study the contact process during docking. Thirdly, we investigate how SMSD design parameters such as volume, moment of inertia, mass, and catenary stiffness affect the UUV docking process, assessing their impact on docking efficiency, accuracy, stability, and safety. Following the principle of a single variable, only the volume changes the physical shape of the suspended body when it varies, while the moment of inertia, mass, and catenary stiffness are treated as parameter variables in the simulation. Finally, we discuss the dynamic characteristics of SMSD, providing insights for the optimization of the design of suspended dock systems and propose strategic suggestions for UUVs.

Table 5. Table of symbols for evaluation indices.

Category	Evaluation Index	Symbol
Attitude of SMSD	Horizontal water resistance of SMSD	$F_{fx}$
	Pitch angle of SMSD	${\psi }_D$
	Variation of the yaw angle of SMSD	${\phi }_D$
	Horizontal displacement of the center of mass of SMSD	$x_D$
	Vertical displacement of the center of mass of SMSD	$z_D$
	Proportion of effective entrance area	$P_{er}$
	Balancing time	$t_{sta}$
Motion response during UUV docking	Docking time	$t_{suc}$
	Angle between the UUV’s central axis and the guiding funnel’s central axis on the horizontal plane	$\gamma$
	Initial contact force	$F_{first}$
	Number of contacts	$\chi$
	Maximum contact force	$F_{max}$
	Roll angle of SMSD	${\vartheta }_D$
	Variation of the yaw angle of the UUV	${\phi }_U$
	Velocity of the UUV’s x axis	$v_x$
	Velocity of the UUV’s y axis	$v_y$
	Angular velocity of the UUV’s z axis	$w_z$

lists the evaluation indices and their definitions to clearly present the numerical simulation results.

4.1. Attitude of SMSD

Regarding attitude stability, the inclination of the dock must be maintained within specific limits under disturbances in water flow to satisfy UUV docking needs. The attitude of the SMSD influences the docking depth of the UUV, the effective entrance area, and its approach heading. This analysis examines two key parameters, namely SMSD mass and catenary stiffness.

4.1.1. Effects of the SMSD Mass

Under constant volume conditions, the mass of the SMSD significantly influences the magnitude of the net buoyancy forces.

Table 6. Simulation results of net buoyancy force of the SMSD under different masses.

${\mathit{m}}_{\mathit{d}}$ (kg)	Mean Net Buoyancy Force (N)
2800	7119
2100	14,029
1400	20,918

displays the mean net buoyancy forces for various masses, which correspond to the theoretical predictions. As depicted in Figure 7a, the increase in $F_{fx}$ corresponds to the empirical formula of $F_f={\displaystyle \frac{1}{2}}\rho B{v_f}^2{C}_d$. As shown in Figure 7b, under the water current, $x_D$ is greater, with a maximum value of 1.536 m, compared to a maximum $z_D$ of 0.33 m. To avoid collision between the UUV and the seabed during docking, the UUV should maintain a minimum safe height of 3.5 m above the seabed. The altitude of the funnel is 4.2 m. With an applied safety factor of 1.4, $z_D$ should remain below 0.5 m to meet the UUV safety standards. As shown in Figure 7c, with a consistent flow velocity, a higher mass results in an increase in ${\psi }_D$, which is particularly noticeable when the mass exceeds 2100 kg. At a flow velocity of 1 m/s and a mass of 2800 kg, ${\psi }_D$ peaks at $20.6^{\circ }$, lowering $P_{er}$ to 87.59% and consequently reducing the apparent area of the guiding funnel opening for the UUV. As observed in Figure 8, the fluid speed on the right side of the SMSD is comparatively low (blue). As the flow velocity increases, the length of the blue region also extends. To ensure stability, ${\psi }_D$ should be kept below ${10}^{\circ }$, resulting in an effective entrance area 97%. To improve the resistance of the SMSD against water flow and reduce its inclination, it is suggested to increase the net buoyancy force. This can be achieved by using materials with a lower density.

The mass of the SMSD has little impact on ${\phi }_D$.

Table 7. Simulation results of the SMSD yaw angle under different flow velocities.

${\mathit{v}}_{\mathit{f}}$ (m/s)	${\mathit{\phi }}_{\mathit{D}}$ (∘)
0.25	oscillate between −3.5 and 4.6
0.5	−8.5
0.75	−8.2
1	−1.1

presents ${\phi }_D$ at different flow velocities. With increasing flow velocity, ${\phi }_D$ decreases, improving downstream performance. When the flow velocity is below 0.25 m/s, ${\phi }_D$ oscillates without converging. To mitigate this problem, it is necessary to incorporate stabilizing fins or a comparable design to facilitate the convergence of ${\phi }_D$ and minimize its magnitude.

4.1.2. Effects of the Catenary Stiffness

Due to the net buoyancy forces, the catenary is under high tension. In Figure 9, the attitude balance curves illustrate the motion characteristics of the SMSD in the vertical and horizontal directions, demonstrating oscillatory, decaying, and stabilizing behaviors. These dynamics are attributed to fluid dynamic damping, which steers the system towards a state of equilibrium. The simulation results indicate that catenary stiffness does not affect the final attitude of the SMSD but does influence the $t_{sta}$. Figure 10 shows a direct proportionality between $t_{sta}$ and catenary stiffness. Consequently, in situations where the strength is satisfied, selecting a mooring chain with lower stiffness is advantageous for the SMSD to reach stability more quickly.

Considering the actual design values for volume, mass, moment of inertia, and catenary stiffness and adhering to the single variable principle, we evaluate the impact of different design parameters.

4.2. Motion Response during UUV Docking Process

Collision is usually the phenomenon of two objects moving relative to each other, touching, and rapidly changing their state of motion. In the docking process, the contact interaction between the UUV and the guiding funnel is crucial to passively adjusting the UUV’s motion, which is a key factor in achieving successful docking. The UUV adopts a sailing docking strategy at a fixed depth/altitude with high-accuracy sensors, such as an altimeter, which makes less vertical deviations. Hence, it can be assumed that the UUV is at the same depth as the center of the guiding funnel. Consequently, our focus is on the docking of the UUV within the horizontal plane. To study the contact process of the UUV docking operation, based on the simulation results reported in reference [33] and our field test experience, the deviation of the horizontal position and the horizontal posture are set at 0.777 m and ${15}^{\circ }$, respectively, as shown in Figure 11. We define successful docking as the vertical channel of the UUV bow entering into the entrance of the SMSD.

With regard to the contact reliability of the SMSD, the emphasis is on its stability and the motion adjustments of the UUV during docking. Figure 12 illustrates the contact forces and kinematic parameters throughout the UUV docking process. The $F_{max}$ of 7894 N does not always occur at the first contact [33], but the initial contact generates dominant torque and rotational energy, driving the movements of the SMSD and UUV. The catenary force stabilizes at −4800 N, with larger contacts causing greater fluctuations before stabilization. Contact forces lead to sudden variations in UUV speed and angular velocity [30], with the axial speed ($v_x$) decreasing and the lateral speeds ($v_y$ and $v_z$) showing lower amplitudes. The $w_z$ remains consistently low in amplitude and decreases further due to the damping effect of water flow. From Figure 12a and Figure 13a, it is apparent that the UUV makes contact with the guiding funnel at 0.324 s, 0.724 s, and 1.231 s, and it can be intuitively seen that the UUV’s turning after contact is relatively slight. This indicates that $w_z$ is small, while $v_x$ is large, resulting in secondary contacts. This occurs because the flexible structure of the SMSD can absorb contact impacts, reducing $w_z$ and contact forces and resulting in a loss of kinetic energy. Compared to docking with a fixed dock, the number of contacts increases when docking with a suspended dock. As shown in Figure 12f and Figure 13, the SMSD attitude angle increases steadily, facilitating the alignment of the guide funnel with the UUV, while ${\phi }_D$ changes less than ${\phi }_U$, providing an adequate reaction force and torque [31].

Therefore, during the UUV docking process, the SMSD maintains a stable attitude under fluid dynamics. As shown in Figure 13, the UUV adjusts its direction through multiple light contacts, improving UUV sailing stability and safety. The SMSD’s flexible mooring reduces contact force magnitudes and the risk of damage. Furthermore, if an incorrect attitude causes the UUV to get stuck, increasing the propeller thrust can rotate the SMSD and correct the attitude angle, contributing to the success of the docking.

4.3. Effects of Various Design Parameters on UUV Docking

The single-chain mooring method improves the flexibility of the dock but requires careful consideration of the design. It should enable sufficient steering adjustments for the UUV and appropriate flexible performance to avoid causing too many contacts or docking failures. This section analyzes the impact of volume, moment of inertia, mass, and catenary stiffness on docking, concluding with suggestions for SMSD optimization.

4.3.1. Effects of the SMSD Volume

The volume of the SMSD significantly affects its buoyancy, water resistance, and damping. As shown in Figure 14, the contact force increases monotonically, while $t_{suc}$, $\chi$, and the attitude angles first decrease, then increase, indicating that volume has a substantial impact. When the volume is less than 1.775 m³, SMSD experiences low water resistance, leading to larger ${\psi }_D$ and ${\phi }_D$ values, which in turn, cause the UUV to pitch, with an increase $\chi$ and the prolongation of $t_{suc}$. For volumes between 1.775 m³ and 14.2 m³, the changes in contact force and $t_{suc}$ are minimal, with decreases in $\chi$ and $\gamma$, resulting in improved SMSD stability and docking accuracy. When the volume reaches 14.2 m³, $\gamma$ is −3.74°, indicating a directional change. This results from a significant $F_{max}$, leading to a pronounced ${\phi }_U$. At a volume of 21.5 m³, $F_{first}$ dramatically increases to 20,372 N, and $\gamma$ increases to 13.65°, increasing the risk of damage and hindering the docking process. As illustrated in Figure 15, excessive contact forces cause the UUV to turn sharply. The UUV repeatedly contacts the opposite side of the guiding funnel, resulting in ${\phi }_U$ fluctuating from 20.18° to 2.53°, then to 7.25°. This leads to instability and difficulty in completing the docking.

The volume of the SMSD significantly affects $t_{suc}$ and $\gamma$. An appropriate volume design can balance docking efficiency, accuracy, and UUV navigation stability. Smaller volumes prolong $t_{suc}$ and destabilize the SMSD’s attitude, while larger volumes increase the contact force, compromising the stability and safety of UUV docking navigation.

4.3.2. Effects of the Moment of Inertia of the SMSD

The moment of inertia of the SMSD influences its resistance to rotation and docking dynamics. As shown in Figure 16, the change in $t_{suc}$ is negligible, but higher moments of inertia increase $F_{max}$ significantly, heightening the risk of UUV damage and, thereby, reducing safety. This is because during the docking process, additional collisions occur nearer to the entrance or central point of the SMSD. Furthermore, a moment of inertia of [5000, 5000, 4500] kg·m² leads to five collisions, three of which show contact forces below 600 N, suggesting minimal impact due to the rectangular bow of the UUV. Furthermore, it can be inferred from $\chi$ and $\gamma$ that a greater moment of inertia enhances the accuracy of UUV docking by better adjusting its motion but reduces ${\phi }_D$ (Figure 16c). Interestingly, ${\phi }_U$ remains stable regardless of SMSD inertia. Optimal docking performance is achieved with inertia moments of [2000, 2000, 1800] kg·m² and [3000, 3000, 2700] kg·m², balancing accuracy, efficiency, and safety.

The moment of inertia of the SMSD significantly affects $F_{max}$, $\gamma$, and ${\phi }_D$. A well-designed moment of inertia ensures the dock’s flexibility and balances docking accuracy and safety. Although a larger moment of inertia improves the accuracy and stability of the docking, it can compromise safety. Thus, an increased moment of inertia is advised while ensuring safety.

4.3.3. Effects of the SMSD Mass

Mass affects the momentum, kinetic energy, and stability of the SMSD. As shown in Figure 17, a mass of 280 kg results in a longer $t_{suc}$ due to the higher pitch angles of the SMSD, increasing the distance from the UUV. $F_{first}$ is sensitive to changes in mass due to net buoyancy, but $F_{max}$ is mainly determined by the contact position and not by mass. $\chi$ and the ${\phi }_D$ show minimal variation. Overall, a mass of 2800 kg offers balanced and optimal performance, while 1400 kg and 2030 kg are less optimal but effective. Larger SMSD masses generally improve docking efficiency but slightly reduce accuracy, mainly affecting $t_{suc}$ and $\gamma$.

4.3.4. Effects of the Catenary Stiffness

Catenary stiffness indicates the flexibility of the mooring chain. As shown in Figure 18, $F_{first}$ generally increases, while $t_{suc}$ and $F_{max}$ do not exhibit significant trends. Other parameters exhibit minimal fluctuations. Catenary stiffness values of 2.59 × 10⁷ N/m and 5.83 × 10⁷ N/m are found to be suitable to adjust the contact force. Selecting the optimal catenary stiffness can reduce the attitude disturbance of the SMSD, ensure smooth docking, and improve docking reliability. In general, this can be achieved by selecting a mooring chain with lower stiffness while still satisfying strength requirements.

4.3.5. Discussion

Comparative analysis of the evaluation indicators reveals the following findings.

• Docking time is primarily influenced by volume and mass.
• Contact force is mainly affected by volume and the moment of inertia.
• The number of contacts is sensitive to volume and mass; it first decreases, then stabilizes as the moment of inertia and catenary stiffness increase.
• The angle between the UUV’s central axis and the guiding funnel’s central axis on the horizontal plane is negatively correlated with volume and the moment of inertia and positively correlated with mass; the influence of catenary stiffness is negligible.
• The variation of the SMSD yaw angle is mainly influenced by volume and the moment of inertia (negatively correlated with both), and volume is satisfied within a specific range of conditions. High volume causes large contact forces, making the SMSD yaw angle unstable.
• The variation of the UUV yaw angle is mostly affected by volume and mass, with a negative correlation with mass.

Therefore, the key parameters influencing the UUV docking effect, in order of significance, are volume, moment of inertia, mass, and catenary stiffness. An optimal volume balances docking efficiency and UUV navigation stability. A larger mass improves docking efficiency and the stability of the suspended dock. An appropriate moment of inertia ensures the flexibility of the suspended dock, balancing docking accuracy and safety. Greater catenary stiffness supports substantial contact forces. When designing a suspended dock, it is crucial to consider the interrelationship of these parameters to ensure efficient, accurate, stable, and safe docking.

5. Sea Experiment

A trade-off is required to choose the optimal combination of design parameters. A design principle can be established by initially determining the volume, then examining the effects of mass and moment of inertia on stability and reliability; choosing a suitable catenary stiffness based on structural strength; and, finally, refining the design optimization. As summarized in

Table 8. Combination of design parameters for the SMSD.

${\mathit{V}}_{\mathit{D}}$ (m3)	${\mathit{m}}_{\mathit{d}}$ (kg)	${\mathit{M}}_{\mathit{I}}$ (kg·m2)	Catenary
			${\mathit{m}}_{\mathit{l}}$(kg)	D (N/m)
3.55	2800	[3000, 3000, 2700]	5.7	2.59 × 107

, we balance the parameters to minimize the docking time while allowing for a moderate increase in maximum contact force within an acceptable range. The angle between the UUV’s central axis and the guiding funnel’s central axis on the horizontal plane, the number of contacts, the UUV yaw angle, and the SMSD yaw angle are kept at moderate levels to ensure comprehensive system performance.

In August 2022, we tested SMSD performance in a UUV docking experiment at sea in Dalian. The depth was 9–10 m, with a current of 1–2 knots. The SMSD deployment process is illustrated in Figure 19. The guide funnel is made up of steel rods. The suspended body is made of carbon steel, and the syntactic foam. The mooring chain is made of stainless-steel material. Given the test conditions, the length of the mooring chain was set at 2 m, and the counterweight was 1.5 tons. Initially, the guide funnel, the suspended body, the mooring chain, and the counterweight were assembled as a single unit (Figure 19a). The SMSD was then deployed with the sling taut under load (Figure 19b). Finally, the counterweight led to the complete submersion of the system. This resulted in the slacking of the sling and a gradual rotation of the SMSD in the current until it reached equilibrium (Figure 19c). The attitude sensor on SMSD showed a stable pitch of 7.5° and a stable roll of 8°. The SMSD takes advantage of the hydrodynamics of the marine environment to maintain stability, thereby enhancing adaptability.

The yaw angle of the SMSD can be calculated on the basis of acoustic measurements [47]. As illustrated in Figure 20, the SMSD maintains a stable orientation of 225°, demonstrating its ability to remain suspended in the presence of current. Furthermore, between 100 s and 160 s, the relative angle between the UUV and the SMSD approaches 0°, indicating that the UUV heads toward the funnel-shaped entrance for docking. The cross-track error decreases to 0 m at 160 s, indicating successful UUV docking. However, after 160 s, there is a fluctuation in the relative angle and the cross-track error. This is due to a contact between the UUV and the guide funnel during docking, causing the SMSD to rotate. Moreover, at the sea test site, it was observed that the proper rotation of the SMSD can facilitate the successful docking of the UUV. The SMSD then returns to its original direction due to a restoring moment of force generated by the current and mooring chain, demonstrating the dynamic stability of the SMSD.

6. Conclusions

This paper uses CFD and DFBI to simulate the stability of an SMSD under ocean currents and to analyze its dynamic response during UUV docking. Furthermore, it offers a detailed evaluation of how critical design parameters such as volume, moment of inertia, mass, and catenary stiffness influence the behavior of the UUV and SMSD during docking. The simulation results improve our understanding of flexible mooring methods and contact dynamics, providing insights into the design of suspended docks, which are listed as follows.

• The pitch of the SMSD should be kept within 10°. Increasing the net buoyancy force and incorporating stabilizer fins can improve SMSD stability and reduce the pitch angle of the SMSD.
• Low stiffness in the mooring chain facilitates the SMSD in quickly reaching equilibrium, resulting in a stable state.
• The SMSD’s mooring connection allows for flexibility, absorbing the contact impacts and resulting in low UUV angular velocity, high UUV velocity, and reduced contact force, lowering damage risk.
• The key parameters affecting UUV docking, in order of impact, are volume, moment of inertia, and mass. Catenary stiffness is negligible.
• Optimal design parameters involve a careful trade-off. Moderate volume balances docking efficiency and UUV navigation. Increased mass enhances both docking efficiency and SMSD stability. A suitable moment of inertia maintains SMSD flexibility, aiding docking accuracy and safety. High catenary stiffness increases contact force.

Finally, after developing the SMSD with a design optimized for stability, sea experiments were conducted. The results revealed that before docking, the SMSD held a stable attitude with a pitch of 7.5°, a roll of 8°, and an azimuth of 225°. During docking, the SMSD rotated but was restored to its initial azimuth after UUV docking. This confirms the stability and contact reliability of the moored suspended structure.

The experimental results show the effectiveness of the proposed SMSD design. However, in weak currents, it shows yaw-angle oscillations that prevent it from stabilizing. Additionally, there is a reduction in docking accuracy, highlighting the need to improve the capability of SMSDs to adjust the UUVs’ steering. Future work will design a “self-guiding wing plate” for the suspended body that adjusts its size based on water flow, optimizing the flow field around the docking station and, thereby, increasing SMSD stability in various currents. Additionally, we consider adding an active control mechanism for the SMSD. Furthermore, the incorporation of a two-way fluid–structure interaction (FSI) approach in the numerical model should be considered, along with a penalty function [48] to refine contact force calculations. This approach will improve the analysis of UUV docking strategies and further improve system performance.